RENOVAL CHARGE TECHNIC APPLIED TO A BIFACIAL SOLAR CELL UNDER CONSTANT MAGNETIC FIELD

An n-p-p bifacial solar cell under constant magnetic field is placed in a fast-switch-interrupted circuit and submitted to a constant multi-spectral illumination. The transient decay occurs between two steady states through operating points depending on two variable resistors; this allows us to obtain a transient decay at any operating point of the I-V curve of the solar cell, from the short circuit to the open one. The influence of magnetic field on the transient photocurrent has been studied using Matlab Simulink simulations. These simulations lead to an equivalent circuit of the bifacial cell in transient state assuming that the photocurrent is the diffused. PACS: 73.50.Pz


INTRODUCTION
This work presents a transient study of a bifacial silicon solar cell under magnetic field and submitted to constant white light (Figure 1). To perform the simulation with Matlab Simulink, the bifacial cell in the experimental setup has been replaced by an equivalent electrical circuit (Mialhe, et al., 1992(Mialhe, et al., , pp 2317, (Barro, et al., 2004) and the influence of the magnetic field on the transient photocurrent has been studied. This equivalent electrical circuit includes the photocurrent source (Colomb, et al., 1992), the device depletion capacitance, the inductance, the series resistance Rs α , and the shunt resistance Rsh α . Rc is the external load.

THEORY
The excess minority carrier concentration is obtained by solving the equation of continuity (Equation 1) with boundary conditions (Equation 2 and 3) which include interface states in the bifacial solar cell.
The boundary conditions are: Sf α is the junction recombination velocity which takes into account the intrinsic recombination velocity Sf 0 and Sb α is the back surface recombination velocity. The steady state excess minority carrier density is: with: ε and γ taking the values in table 1.

Table1. Range of ε and γ
Illumination mode ε γ Front side illumination (α = fr) 1 0 Rear side illumination (α = re) 0 1 Simultaneous illumination (α = s) 1 1 K i is given by : where a i and b i are coefficients deduced from modelling of the generation rate overall radiations in the solar spectrum (SZE, 1981, p.145) The subscript α indicates how the solar cell is illuminated: front side (α = fr), rear side (α = re) or simultaneous illumination (α = s).
Constants A α and B α are determined by means of boundary conditions (Equation 2) and (Equation 3) where the minority carrier density is replaced with (Equation 4).
(Equation 1) and boundary conditions (Equation 2 and 3) constitute a typical Sturm-Liouville's system (BASU and SINGH, 1994, pp.317-329) from which we obtain the minority carriers density in transient state: X α,n and T α,n (0) are deduced from δ α (x,t) by normalization and Fourier transform. τ c,α,n is the decay time constant given by : where ω α,n is the eigenvalue of the transcendental equation:

SIMULATION
To perform the simulation with Matlab Simulink, the bifacial cell in the original setup has been replaced by an equivalent electrical circuit, Figure2, (Barro, et al., 2004) and the influence of the magnetic field on the transient photocurrent has been studied. The Simulink model of the Solar cell equivalent Circuit is shown in Figure 3.  When the time, t < 0 the solar cell is in parallel with the resistor R2 giving the potential V2 corresponding to the operating point F2. At time t=0, the fast switch begins turning on and at time t>0 is completely turned on. It then connects the resistor R2 in parallel with the solar cell and the resistor R1; The voltage V2 drops from V2 to V1 corresponding to the new operating point F1 as it can be seen on Figure 4.   The transient photocurrent decreases if the magnetic field increases for each illumination. Based on Figure 8 we can write the following equation: The series resistance Rs α and the capacitance C α depend on the magnetic field. (Sissoko et al., 1998(Sissoko et al., , pp.1852(Sissoko et al., -1855, (Shockley, 1949, p 435), (Muller and Kamins,1986), (Bierhals, et al., 2000, pp142-145), (Ricaud,1997)  ( 1 1 ) We only investigate solutions corresponding to Δ>0 and Δ<0. When Δ>0 the solution can be expressed by: and R 0 = Rc-Rs

RENOVAL CHARGE TECHNIC APPLIED TO A BIFACIAL SOLAR CELL UNDER CONSTANT MAGNETIC FIELD
Constants A1 and A2 can be calculated for two points : at the maximum current (dI /dt = 0, beginning of the decay) and when the current I is approximately constant corresponding to the end of the decay ( greater value of time, see for example Figure 6 ) I(t)=Is at t= ts.
Constants A 1 , A 2 and K depend on C α , Rsα, sdω α,0 , ts, τ α , L H and Rc. L H and τ α are fixed at 0.001H and 4.5μs respectively. Based on the relation V α (t)= Rc.I α (t), we plotted the photovoltage V α versus the time t for each illumination mode.  10 and 11 show that under the influence of the magnetic field the photovoltage decay increases when the solar cell is illuminated by front side, rear side and both front side and back-side. In the case, Δ <0 the solution can be expressed by:  When Δ is negative, the photovoltage is sinusoidal for each illumination (Figure 12 to 14) and we notice three parts: the first is where the photovoltage increases with the magnetic field, in the second the transient photovoltage doesn't follow the increase of the magnetic field and the last part is where the photovoltage is approximately constant for the three magnetic field. Thus, the case Δ>0 is more interesting than the case Δ<0. The exponential fit of the transient photovoltage when Δ>0 can give values of b, c and a. Value of Rc is fixed, Rs α is negligible compared to Rc for front side illumination and for simultaneous illumination of both front and back-side; b and c can be taken as a system of equations which system lead to the values of L H and C α for these two illumination modes. With the value of a the lifetime τ α is calculated assuming that the eigenvalue of the fundamental mode, ω α,0 is known (Barro, et al., 2004).

CONCLUSION
A theoretical study of a bifacial solar cell under constant magnetic field and under constant illumination was presented. The effect of the magnetic field has been exhibited on both photocurrent and photovoltage for each illumination mode. Based on theoretical and simulation results, the electrical parameters L H , C α , R S and the minority carriers' lifetime τ α are determined for each illumination mode.